Investigating Two Fp Groups of A. Cavicchioli

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This is a page on GAP 3, which is still available, but no longer supported. The present version is GAP 4 (See Status of GAP 3).

Investigating Two Fp Groups of A. Cavicchioli

We first investigate the group h with the parameter q=1.

We define the fp group h and its subgroup s generated by the first generator.

  gap> f := FreeGroup(2); a:=f.1;; b:=f.2;;
  Group( f.1, f.2 )
  gap> h := f/[a*b*a^-2*b*a*b^-1,(b^-1*a^3*b^-1*a^-3)^1*a];
  Group( f.1, f.2 )
  gap> s := Subgroup(h,[h.1]);
  Subgroup( Group( f.1, f.2 ), [ f.1 ] )

In order to see the working of the Todd-Coxeter method determining the index of s in h we set a suitable print level.

  gap> InfoFpGroup1:=Print;; InfoFpGroup2:=Print;;
  gap> Index(h,s);
  #I  CosetTableFpGroup called:
  #I      defined deleted alive   maximal
  #I	  13	  1	  12	  13
  12

We determine the permutation representation of h on the cosets of s and the order of that permutation group.

  gap> p := OperationCosetsFpGroup(h,s);
  Group( ( 2, 3, 5, 7, 9)( 4, 6, 8,10,11), ( 1, 2, 4, 6, 5)( 7, 9,11,12, 8) )
  gap> Size(p);
  60

Encouraged by the success we now look at the regular permutation representation of h.

  gap> Index(h,TrivialSubgroup(h));
  #I  CosetTableFpGroup called:
  #I      defined deleted alive   maximal
  #I	  133	  13	  120	  133
  120

Next we want to learn more about the structure of the group s the order of which we know to be 120.

  gap> AbelianInvariants(h);
  [  ]

So the group h is perfect and, in fact, isomorphic to SL(2,5).

We next try the same procedure with the group g with the parameter q=2.

  gap> g := f/[a*b*a^-2*b*a*b^-1,(b^-1*a^3*b^-1*a^-3)^2*a];
  Group( f.1, f.2 )
  gap> Index(g,Subgroup(g,[g.1]));

In the following output describing the performance of the Todd-Coxeter method, which in this case does not succeed within the maximally allowed 64000 coset numbers, we leave out several lines in the printing below.

  #I  CosetTableFpGroup called:
  #I      defined deleted alive   maximal
  #I	  999	  1	  998	  999
  #I	  1997	  3	  1994	  1996
  . . .

  #I	  62929	  71	  62858	  62889
  #I	  63927	  73	  63854	  63887
  Error, the coset enumeration has defined more than 64000 cosets:
  type 'return;' if you want to continue with a new limit of 128000 cosets,
  type 'quit;' if you want to quit the coset enumeration,
  type 'maxlimit := 0; return;' in order to continue without a limit,
   in
  CosetTableFpGroup( G, H ) called from
  G.operations.Index( G, U ) called from
  Index( g, Subgroup( g, [ g.1 ] ) ) called from
  main loop
  brk> quit;

Rather than continuing the enumeration with an increased limit we have quitted it in order to try a different approach. We can now reset the print level to reduce the output.

  gap> InfoFpGroup1:=Ignore;; InfoFpGroup2:=Ignore;;

We first check that the group g is perfect.

  gap> AbelianInvariants(g);
  [  ]

Next we determine all subgroups of g of index less than or equal to 15 and identify the type of the transitive permutation groups acting on their cosets.

  gap> u := LowIndexSubgroupsFpGroup(g,TrivialSubgroup(g),15);;
  gap> u := Filtered(u,i->Index(g,i)>1);;
  gap> list := List(u,i->OperationCosetsFpGroup(g,i));;
  gap> Length(list);
  24
  gap> id := List([1..24],
  > i->[Index(g,u[i]),Size(list[i]),TransitiveIdentification(list[i])]);;
  gap> Collected(id);
  [ [ [ 11, 660, 5 ], 2 ], [ [ 11, 19958400, 7 ], 8 ], [ [ 12, 660, 179 ], 1 ], 
    [ [ 12, 95040, 295 ], 4 ], [ [ 12, 239500800, 300 ], 3 ], 
    [ [ 13, 3113510400, 8 ], 6 ] ]

We ensure that all these permutation groups are simple.

  gap> Collected(List(list,i->IsSimple(i)));
  [ [ true, 24 ] ]

From the orders we therefore see that the occurring groups are isomorphic to the alternating groups of degrees 11, 12, and 13, the group PSL(2,11), and the Mathieu group M₁₂.
We now form the subdirect product sd of the permutation groups obtained above and verify that this subdirect product is indeed a homomorphic image of our given group g.

  gap> d := DirectProduct(list);;
  gap> Sum(id,i->i[1]);
  284
  gap> sd := Subgroup(d,[
  > Product([1..24],i->d.generators[2*i-1]),
  > Product([1..24],i->d.generators[2*i])]);;
  gap> List(g.relators,i->MappedWord(i,g.generators,sd.generators));
  [ (), () ]

The order of the subdirect product is huge.

  gap> Size(sd);
  187838840781137083803280175646550961040094223793955611966199644315614527534543\
  5097975617726355751505731109984424427520000000000000000000000000000000000000

We hence suspect that the group g may be infinite. It is perfect, but there is a chance that some of its subgroups may be not. So we determine the types of the commutator factor groups of the subgroups we had found before.

  gap> List(u,i->AbelianInvariants(i));
  [ [ 11, 13 ], [ 2, 3, 9 ], [ 11, 13 ], [ 11, 13 ], [ 11, 13 ], [ 0, 0, 2 ], 
    [ 2, 2, 8 ], [ 2, 3, 9 ], [ 11, 13 ], [ 2, 2, 8 ], [ 11, 13 ], [ 2, 2, 8 ], 
    [ 2, 2, 8 ], [ 2, 2, 8 ], [ 2, 2, 3 ], [ 2, 2, 3 ], [ 2, 2, 3 ], 
    [ 2, 2, 3 ], [ 2, 2, 3 ], [ 2, 2, 3 ], [ 2, 2, 3 ], [ 2, 2, 3 ], 
    [ 2, 2, 3 ], [ 2, 2, 3 ] ]
  gap> quit;

The sixth of these types is [ 0, 0, 2 ], telling us that indeed the corresponding subgroup has an abelian factor group that is a direct product of a free abelian group of rank 2 times a cyclic group of order 2.